Causal Model

Model Details

A Bayesian Model for Time-Varying Parameters

A piecewise exponential model for lifetimes.

• Given set \(\mathcal{S}=\left\{s_{0}=0, s_{1}, \ldots, s_{J}\right\}, s_{0}<s_{1}<\cdots<s_{J}\) partitions the time axis into \(J\) intervals \(\left(s_{0}, s_{1}\right], \ldots,\left(s_{J-1}, s_{J}\right]\)

• Hazard within interval is constant

\[ \lambda(t|\boldsymbol z_i; t \in (s_{j-1}, s_j]) = \lambda_{ij} = \exp\left(\beta_{0j} + \sum_{k=1}^{K} z_{i k} \beta_{kj}\right) \]

Piecewise Exponential Model

Evolution of the \(\beta_{kj}\)’s

As in Hemming and Shaw (2002), Gaussian random walks with initial state \(\beta_{k 0} \sim \mathcal{N}\left({\beta_{k}}, {\theta_{k}}\right)\) are considered: \[ \beta_{k j}=\beta_{k, j-1}+w_{j}, \quad w_{j} \sim \mathcal{N}\left(0, {\theta_{k}}\right). \]

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Priors on Innovation Variances and Initial Value Means

Triple gamma priors (Cadonna, Frühwirth-Schnatter, and Knaus 2020)¹ are placed on both \(\beta_k\) and \(\theta_k\). Name stems from the fact that, when used for variances, it has a representation as a compound distribution consisting of three gamma distributions:

\[ \begin{aligned} \theta_{k}\mid{\xi}_{k}^{2} \sim \mathcal{G}\left(\frac{1}{2}, \frac{1}{2 \xi_{k}^{2}}\right), \quad& \xi_{k}^{2}\mid a^{\xi}, \kappa_{k}^{2} \sim \mathcal{G}\left(a^{\xi}, \frac{a^{\xi} \kappa_{k}^{2}}{2}\right), \\ \kappa_{k}^{2} \mid c^{\xi}, \kappa_{B}^{2} &\sim \mathcal{G}\left(c^{\xi}, \frac{c^{\xi}}{\kappa_{B}^{2}}\right). \end{aligned} \]

The first stage conditional prior implies the following first stage conditional prior on \(\sqrt \theta_k\): \[ \sqrt \theta_k | \xi_k^2\sim \mathcal{N}\left(0, \xi_k^2\right) \]

Adding a Factor (?)

To account for unobserved heterogeneity in the data, a grouped factor component can be added to the hazard rates. Let observation \(i\) belong to group \(g\), with \(g \in\{1, \ldots, G\} .\) Then the hazard rates look as follows: \[ \lambda_{i j}=\exp \left(\phi_{g} f_{j}+\beta_{0 j}+\sum_{k=1}^{K} z_{i k} \beta_{k j}\right), \] where \(f_{j}\) is allowed to vary over time according to a zero-mean stochastic volatility law of motion¹: \[ \begin{aligned} f_{j} & \sim \mathcal{N}\left(0, e^{h_{j}}\right), \\ h_{j} \mid h_{j-1}, \phi_{f}, \sigma_{f}^{2} & \sim \mathcal{N}\left(\phi_{f} h_{j-1}, \sigma_{f}^{2}\right),\\ h_{0} & \sim \mathcal{N}\left(0, \sigma_{f}^{2} /\left(1-\phi_{f}^{2}\right)\right) . \end{aligned} \]

Results I

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Results II

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Results III

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Results IV

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Conclusion